C₃₁×#2031 #16841964

order	3968
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₁×#2031

Subgroups

	subgroup	G/H
center	C₄×C₃₁	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂×C₃₁	#393
prime radical	C₂×#41×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

244 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 496

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,33)(34,41,43,47)(35,39)(36,45)(37,40)(38,44,46,42);(34,42,43,44)(38,41,46,47);(32,34)(33,38)(35,42)(36,44)(37,43)(39,47)(40,46)(41,45)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#2031, C₃₁×#23472, C₃₁×#23833, …
maximal normal subgroup of	C₂×C₃₁×#2031, C₃₁×#23472, C₃₁×#23833, …
maximal quotient of	C₃₁×#10362, C₃₁×#10363, C₃₁×#10364, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 763);